# NAG CL Interfaceg13nbc (cp_​pelt_​user)

## 1Purpose

g13nbc detects change points in a univariate time series, that is, the time points at which some feature of the data, for example the mean, changes. Change points are detected using the PELT (Pruned Exact Linear Time) algorithm and a user-supplied cost function.

## 2Specification

 #include
void  g13nbc (Integer n, double beta, Integer minss, double k,
 void (*costfn)(Integer ts, Integer nr, const Integer r[], double c[], Nag_Comm *comm, Integer *info),
Integer *ntau, Integer tau[], Nag_Comm *comm, NagError *fail)
The function may be called by the names: g13nbc or nag_tsa_cp_pelt_user.

## 3Description

Let ${y}_{1:n}=\left\{{y}_{j}:j=1,2,\dots ,n\right\}$ denote a series of data and $\tau =\left\{{\tau }_{i}:i=1,2,\dots ,m\right\}$ denote a set of $m$ ordered (strictly monotonic increasing) indices known as change points with $1\le {\tau }_{i}\le n$ and ${\tau }_{m}=n$. For ease of notation we also define ${\tau }_{0}=0$. The $m$ change points, $\tau$, split the data into $m$ segments, with the $i$th segment being of length ${n}_{i}$ and containing ${y}_{{\tau }_{i-1}+1:{\tau }_{i}}$.
Given a user-supplied cost function, $C\left({y}_{{\tau }_{i-1}+1:{\tau }_{i}}\right)$ g13nbc solves
 $minimize m,τ ∑ i=1 m Cyτi-1+1:τi + β$ (1)
where $\beta$ is a penalty term used to control the number of change points. This minimization is performed using the PELT algorithm of Killick et al. (2012). The PELT algorithm is guaranteed to return the optimal solution to (1) if there exists a constant $K$ such that
 $C y u+1 : v + C y v+1 : w + K ≤ C y u+1 : w$ (2)
for all $u
Chen J and Gupta A K (2010) Parametric Statistical Change Point Analysis With Applications to Genetics Medicine and Finance Second Edition Birkhäuser
Killick R, Fearnhead P and Eckely I A (2012) Optimal detection of changepoints with a linear computational cost Journal of the American Statistical Association 107:500 1590–1598

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the length of the time series.
Constraint: ${\mathbf{n}}\ge 2$.
2: $\mathbf{beta}$double Input
On entry: $\beta$, the penalty term.
There are a number of standard ways of setting $\beta$, including:
SIC or BIC
$\beta =p×\mathrm{log}\left(n\right)$
AIC
$\beta =2p$
Hannan-Quinn
$\beta =2p×\mathrm{log}\left(\mathrm{log}\left(n\right)\right)$
where $p$ is the number of parameters being treated as estimated in each segment. The value of $p$ will depend on the cost function being used.
If no penalty is required then set $\beta =0$. Generally, the smaller the value of $\beta$ the larger the number of suggested change points.
3: $\mathbf{minss}$Integer Input
On entry: the minimum distance between two change points, that is ${\tau }_{i}-{\tau }_{i-1}\ge {\mathbf{minss}}$.
Constraint: ${\mathbf{minss}}\ge 2$.
4: $\mathbf{k}$double Input
On entry: $K$, the constant value that satisfies equation (2). If $K$ exists, it is unlikely to be unique in such cases, it is recommened that the largest value of $K$, that satisfies equation (2), is chosen. No check is made that $K$ is the correct value for the supplied cost function.
5: $\mathbf{costfn}$function, supplied by the user External Function
The cost function, $C$. costfn must calculate a vector of costs for a number of segments.
The specification of costfn is:
 void costfn (Integer ts, Integer nr, const Integer r[], double c[], Nag_Comm *comm, Integer *info)
1: $\mathbf{ts}$Integer Input
On entry: a reference time point.
2: $\mathbf{nr}$Integer Input
On entry: number of segments being considered.
3: $\mathbf{r}\left[\mathit{dim}\right]$const Integer Input
On entry: time points which, along with ts, define the segments being considered, $0\le {\mathbf{r}}\left[i-1\right]\le n$ for $i=1,2,\dots {\mathbf{nr}}$.
4: $\mathbf{c}\left[\mathit{dim}\right]$double Output
On exit: the cost function, $C$, with
 $c[i-1]= Cyri:t ​ if ​t>ri, Cyt:ri ​ otherwise.$
where $t={\mathbf{ts}}$ and ${r}_{i}={\mathbf{r}}\left[i-1\right]$.
It should be noted that if $t>{r}_{i}$ for any value of $i$ then it will be true for all values of $i$. Therefore the inequality need only be tested once per call to costfn.
5: $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to costfn.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling g13nbc you may allocate memory and initialize these pointers with various quantities for use by costfn when called from g13nbc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
6: $\mathbf{info}$Integer * Input/Output
On entry: ${\mathbf{info}}=0$.
On exit: set info to a nonzero value if you wish g13nbc to terminate with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
Note: costfn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by g13nbc. If your code inadvertently does return any NaNs or infinities, g13nbc is likely to produce unexpected results.
6: $\mathbf{ntau}$Integer * Output
On exit: $m$, the number of change points detected.
7: $\mathbf{tau}\left[{\mathbf{n}}\right]$Integer Output
On exit: the first $m$ elements of tau hold the location of the change points. The $i$th segment is defined by ${y}_{\left({\tau }_{i-1}+1\right)}$ to ${y}_{{\tau }_{i}}$, where ${\tau }_{0}=0$ and ${\tau }_{i}={\mathbf{tau}}\left[i-1\right],1\le i\le m$.
The remainder of tau is used as workspace.
8: $\mathbf{comm}$Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
9: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{minss}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{minss}}\ge 2$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 2$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_USER_STOP
User requested termination.

Not applicable.

## 8Parallelism and Performance

g13nbc is not threaded in any implementation.

g13nac performs the same calculations for a cost function selected from a provided set of cost functions. If the required cost function belongs to this provided set then g13nac can be used without the need to provide a cost function routine.

## 10Example

This example identifies changes in the scale parameter, under the assumption that the data has a gamma distribution, for a simulated dataset with $100$ observations. A penalty, $\beta$ of $3.6$ is used and the minimum segment size is set to $3$. The shape parameter is fixed at $2.1$ across the whole input series.
The cost function used is
 $Cyτi-1+1:τi = 2⁢ a⁢ ni log⁡Si - log a⁢ ni$
where $a$ is a shape parameter that is fixed for all segments and ${n}_{i}={\tau }_{i}-{\tau }_{i-1}+1$.

### 10.1Program Text

Program Text (g13nbce.c)

### 10.2Program Data

Program Data (g13nbce.d)

### 10.3Program Results

Program Results (g13nbce.r)
This example plot shows the original data series and the estimated change points.